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Say I have some projective space $\mathbb{P}^n$ and some line bundle $L=\mathcal{O}(-k)$. Now, I want to have a subvariety $Y$ in $\mathbb{P}^n$ such that $L\vert_Y$ is trivial.

When is this the case? I can only think of trivial solutions, like when $Y$ is just a point and I can't seem to find a standard treatment of this in literature

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up vote 8 down vote accepted

Let's exclude the other trivial solution: $k=0$ and any $Y$.

Suppose now that $k\ne 0$. As $O(-k)|_Y$ trivial is equivalent to $O(k)|_Y$ (isomorphic to the dual of $O(-k)|_Y$) trivial, we can restrict to the case $k<0$. Then $L$, hence $L|_Y$, are ample. If $L|_Y$ is moreover trivial, then $O_Y$ is ample, which implies that $Y$ is affine. So necessarily $Y$ is affine. If $Y$ is a closed subvariety, this forces $Y$ to be a finite set.

Conclusion: if $Y$ is a closed subvariety such that $L|_Y$ is trivial with $k\ne 0$, then $Y$ is a finite subset.

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